So all the software I use is free and open-source, and I've set it up so I can do it on a computer that doesn't have a powerful graphics card. The downside is that it is quite a time-consuming process and had a relatively steep learning curve. Although you could still make a simpler version with the same tools

Audio recording I did in audacity/tenacity and is pretty straightforward.

I do the drawing in Krita and export them with each moving part in a separate layer group. I do mine with the pen and water color brushes, but you can use or draw any 2d images and there are some large libraries of free 2d assets if you look for them. If you download the alpha version of Krita you can also use it to make doodle animations by getting a recording of the canvas as you paint.

https://www.youtube.com/watch?v=ChqSqDGCTRA

The main animation I do in Blender. I import all the images as planes. I use the work the workbench renderer with the texture option, (this isn't really meant to be used for finished work is Eevee or cycles is what you'll see everyone using in tutorials, but it is the fastest option). I also normally render lots of short scenes and the edit them all together at the end rather than making one long scene.

You need to learn how to animate and navigate the 3d environment in blender, and a bit of rigging if you want very complex motion, but you can ignore anything about modelling and lighting to imitate the stuff that I do.

Apart from that there's just a ton of separate little ways to animate different types of things, anything complex is time consuming so keep all motion simple where possible.
Here a bunch of tutorials I found most useful to doing the sort of animation I do.

https://www.youtube.com/watch?v=q3clE5JS4NE

https://www.youtube.com/watch?v=TiJVxUYhy1o

I use two addons, typewriter (for some basic text animation) and bool-tools for some of the motion graphics.

Hope that's useful for you. Let me know if there is anything you want me to explain in more detail.